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136formulaFinalS10

136formulaFinalS10 - Math 136 Fall 2010 Formula Sheet for...

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Math 136, Fall 2010, Formula Sheet for Final Exam sin(0) = 0 ; sin( π/ 6) = 1 / 2 ; sin( π/ 4) = 2 / 2 ; sin( π/ 3) = 3 / 2 ; sin( π/ 2) = 1 cos(0) = 1 ; cos( π/ 6) = 3 / 2 ; cos( π/ 4) = 2 / 2 ; cos( π/ 3) = 1 / 2 ; cos( π/ 2) = 0 cos 2 x + sin 2 x = 1 ; 1 + tan 2 x = sec 2 x ; 1 + cot 2 x = csc 2 x sin(2 x ) = 2sin x cos x ; cos(2 x ) = cos 2 x - sin 2 x cos 2 x = 1 2 (1 + cos(2 x )) ; sin 2 x = 1 2 (1 - cos(2 x )) sin A cos B = 1 2 [sin( A - B ) + sin( A + B )] sin A sin B = 1 2 [cos( A - B ) - cos( A + B )] cos A cos B = 1 2 [cos( A - B ) + cos( A + B )] Z sec xdx = ln | sec x + tan x | + C ; Z csc xdx = ln | csc x - cot x | + C If T N , S N are the Trapezoidal and Simpson’s approximations, then T N = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ··· + 2 f ( x N - 2 ) + 2 f ( x N - 1 ) + f ( x N )] S N = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ··· + 2 f ( x N - 2 ) + 4 f ( x N - 1 ) + f ( x N )] . If I = R b a f ( x ) dx then: | T N - I | ≤ K 2 ( b - a ) 3 12 N 2 ; | S N - I | ≤ K 4 ( b - a ) 5 180 N 4 . If f ( t ) is the amount of money deposited at time t over the time period [0 ,T ] in an account that earns interest at the annual rate r compounded continuously, then Future Value = Z T 0 f ( t ) e r ( T - t ) dt Present Value = Z T 0 f ( t ) e - rt dt If q 0 units of a commodity are sold at a price of
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